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Properties

Label 2-234-117.110-c1-0-10
Degree $2$
Conductor $234$
Sign $-0.968 + 0.250i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (−1.53 + 0.805i)3-s + (−0.866 − 0.499i)4-s + (0.0908 − 0.339i)5-s + (0.380 + 1.68i)6-s + (−2.17 − 2.17i)7-s + (−0.707 + 0.707i)8-s + (1.70 − 2.46i)9-s + (−0.304 − 0.175i)10-s + (−3.21 − 0.861i)11-s + (1.73 + 0.0694i)12-s + (−3.46 + 1.01i)13-s + (−2.66 + 1.54i)14-s + (0.133 + 0.593i)15-s + (0.500 + 0.866i)16-s + (−3.87 − 6.71i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.885 + 0.464i)3-s + (−0.433 − 0.249i)4-s + (0.0406 − 0.151i)5-s + (0.155 + 0.689i)6-s + (−0.823 − 0.823i)7-s + (−0.249 + 0.249i)8-s + (0.567 − 0.823i)9-s + (−0.0961 − 0.0555i)10-s + (−0.969 − 0.259i)11-s + (0.499 + 0.0200i)12-s + (−0.959 + 0.280i)13-s + (−0.713 + 0.411i)14-s + (0.0345 + 0.153i)15-s + (0.125 + 0.216i)16-s + (−0.940 − 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0577001 - 0.453410i\)
\(L(\frac12)\) \(\approx\) \(0.0577001 - 0.453410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (1.53 - 0.805i)T \)
13 \( 1 + (3.46 - 1.01i)T \)
good5 \( 1 + (-0.0908 + 0.339i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.17 + 2.17i)T + 7iT^{2} \)
11 \( 1 + (3.21 + 0.861i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.87 + 6.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.67 - 0.984i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.68T + 23T^{2} \)
29 \( 1 + (-4.92 + 2.84i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.09 + 0.293i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-8.90 + 2.38i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.85 - 1.85i)T + 41iT^{2} \)
43 \( 1 - 4.25iT - 43T^{2} \)
47 \( 1 + (1.11 + 4.17i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (2.84 + 10.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (-9.74 + 9.74i)T - 67iT^{2} \)
71 \( 1 + (-0.343 + 1.28i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.19 + 2.19i)T + 73iT^{2} \)
79 \( 1 + (6.22 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.92 - 1.05i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.43 + 5.37i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.597 - 0.597i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.65388384462910054502676157639, −10.83539643711936159146526759516, −9.857039171282868235907801828054, −9.450776558568173563823150572321, −7.61166721986319979176218883970, −6.49003226617479867711003894063, −5.19036302677850352547905541732, −4.33130779611399559365104687986, −2.88142240449885549653896644532, −0.37175040777953967894246225027, 2.56992559898895127186086866337, 4.56188412274959010764730589723, 5.68761241733881040930698446909, 6.39102013903707035971929502639, 7.44220312127265939306519397544, 8.450216776399745609569968243893, 9.835656262848572967194541032267, 10.62152506344898350080913765187, 12.01034483457528442920550705168, 12.66206429159755674664172860425

Graph of the $Z$-function along the critical line